More effective heuristics for a two-machine no-wait flowshop to minimize maximum lateness

We address a manufacturing environment with the no-wait constraint which is common in industries such as metal, plastic, and semiconductor. Setup times are modelled as uncertain with the objective of minimizing maximum lateness which is an important performance measure for customer satisfaction. This problem has been addressed in scheduling literature for the two-machine no-wait flowshop where dominance relations were presented. Recently, another dominance relation was presented and shown to be about 90% more efficient than the earlier ones. In the current paper, we propose two new dominance relations, which are less restrictive than the earlier ones in the literature. The new dominance relations are shown to be 140% more efficient than the most recent one in the literature. As the level of uncertainty increases, the newly proposed dominance relation performs better, which is another strength of the newly proposed dominance relation. Moreover, we also propose constructive heuristics and show that the best of the newly proposed heuristics is 95% more efficient than the existing one in the literature under the same CPU time.


Introduction
The no-wait flowshop is a type of flowshop where consecutive operations are carried out with no delay. This is vital for certain settings in which waiting causes undue difficulty in manufacturing. A typical example is when temperature is involved, and operations must be completed while the material is still hot. Reducing work-in-process is another advantage of the no-wait flowshop, Macchiaroli et al. (1999). Scheduling of patients (Hsu et al., 2003), aircraft landing , trains (Liu and Kozan, 2011), and bakery production (Hecker et al., 2014) are some examples where such a flowshop would be necessary. It is also crucial in the pharmaceutical industry, chemical industry, and plastic industry, Allahverdi (2016), and Hall and Sriskandarajah (1996).
The amount of time required to set up a resource for production is called setup time. Since setup times clearly affect the completion time, it is necessary to consider them for scheduling problems. Nonetheless, a surprising percentage of scheduling research (at least 90%) ignores setup times, Allahverdi (2015). Among those that do not ignore, setup times are considered as known values, e.g., Dileepan (2004), yet there are many manufacturing settings when this is not the case. In fact, far from being known, they may change due to many factors such as unforeseen breakdowns and shortage of equipment, Kim and Bobrowski (1997). For such manufacturing settings, ignoring set up times or considering them as known values will greatly impact the maximum lateness, resulting in an unnecessarily poor efficiency.
In the scheduling literature, the assumption of a deterministic environment (where setup and/or processing times are fixed and known in advance) is commonly utilized, Seidgar et al. (2014) and Keshavarz and Salmasi (2013). On the other hand, manufacturing environments in industries are frequently subject to a wide range of uncertainties, Allahverdi (2022a), Wang and Choi (2012), Gonzalez-Neira et al. (2017). Hence, managers face substantial uncertainty in setup times. Moreover, assuming certain probability distributions for setup times may not be valid for some environments, e.g., Kouvelis and Yu (1997). Thus, it is essential to model setup times as uncertain.
Let UBsi,k and LBsi,k be the upper and lower bounds of setup time si,k of job i on machine k, respectively. Dominance relations are provided by Allahverdi et al. (2003) and Allahverdi (2005) for the problems F2|LBsi,k≤ si,k≤ UBsi,k|Cmax,∑Cj and F2|LBsi,k≤ si,k≤ UBsi,k|Cmax,respectively, where Cmax and ∑Cj symbolize the makespan and completion time. Additionally, dominance relations are provided for the problem F2|LBsi,k≤ si,k≤ UBsi,k|Cmax by Aydilek et al. (2013) assuming fixed processing times and by Aydilek et al. (2015) relaxing the assumption of fixed processing times, i.e., processing times are modelled as uncertain. Allahverdi (2022b) provided and algorithm for the problem of LBsi,k≤si,k≤UBsi,k|Lmax and showed that the algorithm performs much better than the earlier existing algorithms. Other papers studying uncertain environments include Braun et al. (2002), Sotskov et al. (2009), Matsveichuk et al. (2011), Sotskov and Lai (2012, and Sotskov and Matsveichuk (2012), where set up/processing times are also assumed to be within certain upper and lower bounds. Allahverdi and Allahverdi (2018) address the problem F2|no-wait,LBsi,k≤si,k≤ UBsi,k|Lmax which aims to minimize maximum lateness assuming setup times to be uncertain. Another dominance relation for the same problem is provided by Allahverdi et al. (2021), which is proven to be of higher efficiency. In the current paper, we present a new dominance relation and demonstrate that it is much more efficient than the one provided by Allahverdi et al. (2021). Moreover, we propose new heuristics for this problem and demonstrate that they are more effective than the previous heuristics in the literature while maintaining the same computational time.
Problem definition is provided in the next section. The new dominance relations are presented in Section 3 while their evaluation is given in Section 4. The new heuristics are described in Section 5 while their performance compared with the recent heuristic in the literature is given in Section 6. Conclusion remarks are presented in Section 7.

Problem Definition
We address the problem of minimizing maximum lateness in a two-machine no-wait flowshop scheduling environment with separate setup times. A typical two-machine no-wait flowshop problem with three jobs can be seen in a Gantt chart in Figure  1. We model setup times as uncertain and bounded, within lower and upper bounds. We assume that all jobs are available at time zero and every job has a positive processing time on each machine. A machine can process at most one job and a job can be processed on at most one machine at any given time. Let sh,m and th,m denote the setup and processing times of job h on machine m (m=1, 2), respectively. Also let the lateness, completion time, and due date of job h be denoted by Lh, Ch and dh, respectively, where Lh = Ch -dh.

More Efficient Dominance Relations
A dominance relation was presented by Allahverdi and Allahverdi (2018) for the problem. Recently, Allahverdi et al. (2021) presented another dominance relation and showed that their dominance relation is more effective than the dominance relation presented by Allahverdi and Allahverdi (2018). In this section, we propose two new dominance relations (Theorems 2 and 4) and show that the new dominance relations are much more efficient than that of Allahverdi et al. (2021). Let π1 be a sequence such that job j, in position τ, immediately precedes job i, i.e., they are adjacent. Also let π2 be another sequence obtained from π1 by interchanging the jobs i and j; job i precedes job j in sequence π2. Moreover, let the bold Sh,k be a random variable denoting the setup time of job h on machine k, which satisfies Then, the lateness of jobs i and j in the two sequences π1 and π2 for jobs in positions τ, τ+1, and τ+2 are given by by condition a) of the theorem. Therefore, it follows from Eq. (10) and Eq. (12) that By Eq. (7) and Eq. (8), and conditions a), b), and c) of the theorem, it follows that It follows from Eq. (13) and Eq. (14) [ ] = ∆ + max , , , + , − [ , ] + , +max , , , + , − , + , +max [ , ] , [ , ] + [ , ] − , + [ , ] + max [ , ] , [ , ]  Proof: The proof directly follows from Lemma 1, Lemma 2, and Theorem 1.
Corollary 1: Let jobs i and j be two adjacent jobs in a given sequence. If the following conditions hold, for a deterministic problem where setup times are known for certain, Proof: Since the problem is deterministic, Eq.
(2) reduces to , = , = , Moreover, since setup times are known values (denoted by sh,k), the last equation is equivalent to Hence, the proof follows from that of Theorem 2.
Theorem 3: Let jobs i and j be two adjacent jobs in a given sequence. If the following conditions hold, a) , ≤ , and + , ≤ Proof: By Eq.
(2) and conditions b) and c) of theorem 3, Eq. (9) reduces to From Eq. (7) and Eq. (8), and the conditions of Theorem 3, we obtain It follows from Eq. (17) Proof: Note that Lemma 2 holds under the conditions of Theorem 3. Therefore, the proof follows from Lemma 1, Lemma 2, and Theorem 3.
Corollary 2: Let jobs i and j be two adjacent jobs in a given sequence. If the following conditions hold, for a deterministic problem where setup times are known for certain, a) , ≤ , and + , ≤ Proof: For the same reasons stated in the proof of Corollary 1, it is true that Thus, the proof follows from that of Theorem 4.

Numerical Results for Dominance Relations
Allahverdi et al. (2021) presented a dominance relation for our problem and showed that their dominance relation was about 90% more efficient than the earlier existing ones in the literature. In this paper, we develop two new dominance relations (Theorems 2 and 4) for the problem and show that our newly developed dominance relations are more efficient than that of Allahverdi et al. (2021). The dominance relation developed in Theorem 2 is denoted by DR2 and the one in Theorem 4 is represented by DR4. It should be noted that DR2 and DR4 are not overlapping since the conditions stated in Theorems 2 and 4 cannot hold at the same time. In other words, these two theorems are concerned with different cases. Therefore, both theorems can be applied for a given sequence. Hence, we denote the combination of DR2 and DR4 by DRC. Let n denote the number of jobs, R denote the range factor and T denote tardiness factor. Also, let Δ denote the uncertainty range between the lower and upper bounds of setup times. For a fair comparison, we use the same computational settings of Allahverdi et al. (2021). Specifically, where LB-Cmax is a lower bound value on makespan.
Hall and Posner (2001) suggest generating processing times from the uniform distribution U(1,100) since its variance is very large. On the other hand, it is common to generate due dates using a uniform distribution between LB-Cmax(1-T-R/2) and LB-Cmax(1-T-R/2) where R and T are usually taken between 0 and 1, e.g., Kim (1993), and Vallada and Ruiz (2010), and LB-Cmax is a lower bound on makespan. The following LB-Cmax is used.
Tables 1-3 present computational results for Δ=5, Δ=10, and Δ=20, respectively. The numbers in the tables are the average of 250 replications.  Tables 1-3 with respect to n, R, T, and Δ values, respectively, where the y-axis denotes the percentage improvement for dominance relations. Fig. 2 shows the percentage improvements of DR2, DR4, and DRC with respect to the number of jobs. It shows that the percentage slightly decreases as the number of jobs increases. However, the percentage of improvement is at least 120 for the DRC. Results with respect to the R values are given in Fig. 3 where the percentage improvements of the dominance relations are independent of the R values. In other words, the improvement of dominance relations is similar for different due date ranges. A similar effect can be seen from Fig. 4 with respect to tardiness factor T. However, the percentage improvement of the dominance relations increases as the Δ (Delta) values increase. In other words, as the uncertainty level increases, the newly developed dominance relations perform better, which is another strength of the newly developed dominance relations.

Heuristics
Allahverdi et al. (2021) recently presented a constructive heuristic, based on lower and upper bounds of setup times, with five versions for the considered problem. They demonstrated that one version of the heuristics, called CH4, performed better than the rest. In this section, we propose new heuristics for the problem and show in the next section that they are much more efficient than that of Allahverdi et al. (2021) with the same computational time. Let π denote a partial sequence of scheduled jobs and σ the set of jobs which are going to be scheduled. Moreover, let A be the set of all jobs and nw the number of jobs to be scheduled. Also, let Lmax (π) represent the maximum lateness of the partial sequence π.

Proposed Heuristics -PH(α, β)
Step 1: Initialize the set of jobs to be scheduled to the set of all the jobs (σ ←A) and the partial sequence π to the empty sequence (π ← Ø) Step 2: Initialize the number of unscheduled jobs to the number of all jobs (nw ← n) Step 3: Start scheduling the first job, js ←1.
Step 7: Schedule the job t1 as the first job (π ← {t1}) Step 8: Take the job t1 out of the set of unscheduled jobs (σ ← σ \ {t1}) Step 9: Reduce the number of unscheduled jobs by 1 (nw ← nw -1) Step 10: Start the scheduling of the next job (js ← js + 1) Step 11: For each job w from σ a) Form a partial sequence πw by scheduling the job w ϵ σ as the job to be processed after the jobs in the sequence π (πw = π U {job w}). In other words, place the job w in position js of πw. b) Compute the maximum lateness, Lmax (πw), of the partial sequence πw as below: % Improvement % Improvement % Improvement % Improvement Step 12: Among the calculated nw values in Step 11, choose the job w * which has the smallest maximum lateness. In other words, Lmax (πw*) = max {Lmax (πw), w ϵ σ } Step 13: Form js sequences (seq1,…,seqjs) by inserting the job w* in all possible positions (positions 1 up to js) and compute Lmax of each sequence. For example, place the job w* in the first position and shift all the jobs in the sequence π by one position to the right to produce seq1. seq1 ← {job w*} U π Similarly, job w* is placed in the 2 nd position of μ and jobs in positions greater than or equal to two are shifted by one position to the right (if j ≥ 2) in seq2.
Step 14: Among the js sequences formed in Step 13, choose the sequence (seqj*) which has the largest maximum completion time.
Step 18: Take the recently scheduled job w* out of the set of unscheduled jobs (σ ← σ \ {w*}).
Step 19: Go to Step 11 if there are jobs waiting to be scheduled (If js < n) Stop if all of the jobs are scheduled (If js = n) For our proposed heuristics, we set the parameter N to 4 after computational experiments. Hence, we set α and β to be integer values between 0 and 4, specifically α ϵ {0, 1, 2, 3, 4} and β ϵ {0, 1, 2, 3, 4}. This results in 25 heuristics. For example, α=2 and β=3 in the proposed heuristic is denoted by PH(2, 3).

Computational Results for Heuristics
We evaluate the performance of the proposed 25 heuristics, PH(0,0) to PH (4,4), and the best existing heuristic CH4 in the literature. Experimental computations are conducted using the software Matlab on a PC with Intel(R) Core(TM) i7-8550U CPU processor of 1.99 GHz with 8 GB RAM. We set the parameters to the values used in Section 4. To compare the performances of the proposed heuristics and the best one in the literature, the realized setup times (between the lower and upper bounds) are necessary. Since the actual setup times are uncertain and only the lower and upper bounds are known, we considered different distributions to generate actual setup times. These distributions, given below, include both skewed (positive and negative) and symmetric distributions. Details and justifications for using these distributions are given in Allahverdi et al. (2021).
It should be noted that 1,200 replications are conducted for each combination of n, R, T, Δ, and distributions. More specifically, n changes from 30 to 70 with the increment of 10, R and T take values of 0.25, 0.50, and 0.75, Δ takes values of 5, 10, and 20. Six different distributions are used for generating realized setup times, i.e., normal distribution, uniform distribution, positive and negative linear distributions, positive and negative exponential distributions. These six distributions are the distributions that were used in Allahverdi et al. (2021). This results in 8100 combinations, and for each combination, 1200 replications are generated. Thus, the number of problems considered are 97,200 problems. The relative error (RE) of the heuristic PH(i, j) and the benchmark heuristic CH4 are computed as: where "X" denotes one of the heuristics of PH(i, j) and CH4, and "best" refers to the best of these heuristics. There are 25 proposed heuristics and the benchmark heuristic CH4, in total 26 heuristics, to compare with each other. The computational experiments were conducted for all the 26 heuristics. However, reporting all the computational results for all the heuristics is cumbersome and infeasible. Therefore, we present the results of the two best-proposed heuristics, PH(4, 3) and PH(5, 4), and the two worst proposed heuristics, PH(1, 1) and PH(2, 2), along with the benchmark heuristic CH4, in total 5 heuristics.
The RE values for these 5 heuristics are presented in Table 4, Table 5, and Table 6, for Δ=5, Δ=10, and Δ=20, respectively. An entry in a table is the average error regarding the considered distributions with 1200 runs. Standard deviations were small as a result of a large number of replications, and hence, they are not reported. The results in Tables 4-6 are summarized in Figs. 6-10. Fig. 6 indicates the RE values of the five heuristics with respect to the number of jobs, n. The figure clearly shows that the proposed heuristics PH(4, 3), PH(5, 4), PH(1, 1) and PH(2, 2) perform much better than the benchmark heuristic of CH4. Moreover, even though the performance of the proposed heuristics gets better as n increases, as can better be seen in Fig. 7, however, that of the benchmark heuristic CH4 deteriorates. It is seen in Figure 6 that the performances of the proposed heuristics PH(4, 3), PH(5,4), PH(1,1) and PH(2, 2) look much better than the heuristic CH4. Furthermore, it is clear from Fig. 7 (excluding CH4) that the difference between the best and the worst proposed heuristics is about 40%. The results for R, T, and Δ values are presented in Figures 8, 9, and 10, respectively, by excluding CH4 for a vivid comparison of the proposed heuristics. Fig. 8 shows that the performances of PH(4, 3) and PH(5, 4) get slightly better as R increases. Moreover, Figure  9 indicates that the performances of the proposed heuristics get better as T increases. As expected, as Δ values increase, the RE values of the proposed heuristics slightly increase. The overall errors of the proposed heuristics PH(4, 3), PH(5,4), PH(1,1), PH(2,2), and the benchmark heuristic CH4 are 1. 38, 1.37, 2.03, 1.90, 27.56, respectively. This shows that the worst of the proposed heuristics, PH(1, 1), reduces the error of the benchmark heuristic (the best existing heuristic in the literature), CH4, by 92%. Moreover, the error of the best proposed heuristic, PH(5,4), is about 32% better than the heuristic PH(1, 1). Finally, it should be noted that the CPU times of the proposed heuristics and that of the benchmark heuristic CH4 were similar, and it was less than one second for the largest problem size.

Concluding Remarks
The two-machine no-wait flowshop scheduling problem is addressed with the objective of minimizing maximum lateness where setup times are modelled as uncertain with given lower and upper bounds. The problem was addressed earlier in the literature and dominance relations along with heuristics were presented. In this paper, we develop new dominance relations and show that they are at least 130% more efficient than the best existing one in the literature. Moreover, we propose new heuristics and show that the worst of the newly proposed heuristics is at least 90% more efficient than the existing one under the same CPU time. Given that the newly developed dominance relations are less restrictive, and the proposed heuristics are more efficient and they are computationally fast, they can easily be implemented for solving real life application problems. The addressed problem can be extended to other performance measures such as the number of tardy jobs, e.g., Aydilek et al. (2017). Furthermore, setup times are considered as sequence independent in the current paper. However, since setup times may be sequence dependent in some real-life applications (Guevara-Guevara et al. 2022), the results of this paper may also be extended to the case of sequence dependent setup times. Yet another extension to the problem is to consider total tardiness performance measure since this is a commonly considered performance measure, e.g., Braga-Santos et al. (20220, Rosssit et el. (2021), González-Neira and Montoya-Torres (2019) and Akande et al. (2018).